After reading this chapter, you should be able to:
- Understand key differences between crisp set and fuzzy set logics
- Calibrate in an informed way the fuzzy-set membership scores for the different conditions
- See the connection between the multidimensional vector space defined by fuzzy-set conditions and a conventional truth table
- Gain a deeper understanding of the fuzzy subset relationship and of how to calculate and evaluate its consistency
- Relate fuzzy subset relations to the concepts of causal sufficiency and necessity
- Understand the different steps in a fuzzy-set analysis, especially the importance of frequency thresholds and consistency thresholds when creating a crisp truth table summarizing the results of multiple fuzzyset analyses
One apparent limitation of the truth table approach is that it is designed for conditions that are simple presence/absence dichotomies (i.e., Boolean or “crisp” sets—see Chapter 3) or multichotomies (mvQCA—see Chapter 4). Many of the conditions that interest social scientists, however, vary by level or degree. For example, while it is clear that some countries are democracies and some are not, there is a broad range of in-between cases. These countries are not fully in the set of democracies, nor are they fully excluded from this set.[Page 88]
Fortunately, there is a well-developed mathematical system for addressing partial membership in sets, fuzzy-set theory (Zadeh, 1965; Klir, Clair, & Yuan, 1997). This chapter first provides a brief introduction to the fuzzy-set approach, building on Ragin (2000). Fuzzy sets are especially powerful because they allow researchers to calibrate partial membership in sets using values in the interval between  (nonmembership) and  (full membership) without abandoning core set theoretic principles such as, for example, the subset relation. As Ragin (2000) demonstrates, the subset relation is central to the analysis of causal complexity.
While fuzzy sets solve the problem of trying to force-fit cases into one of two categories (membership versus nonmembership in a set) or into one of three or four categories1 (mvQCA), they are not well suited for conventional truth table analysis. With fuzzy sets, there is no simple way to sort cases according to the combinations of conditions they display because each case's array of membership scores may be unique. Ragin (2000) circumvents this limitation by developing an algorithm for analyzing configurations of fuzzy-set memberships that bypasses truth table analysis altogether. While this algorithm remains true to fuzzy-set theory through its use of the containment (or inclusion) rule, it forfeits many of the analytic strengths that follow from analyzing evidence in terms of truth tables. For example, truth tables are very useful for investigating “limited diversity” and the consequences of different “simplifying assumptions” that follow from using different subsets of “logical remainders” to reduce complexity (see Chapters 3 and 4, and also Ragin, 1987, 2008; Ragin & Sonnett, 2004). Analyses of this type are difficult when not using truth tables as the starting point.
A further section of this chapter thus builds a bridge between fuzzy sets and truth tables, demonstrating how to construct a conventional Boolean truth table from fuzzy-set data. It is important to point out that this new technique takes full advantage of the gradations in set membership central to the constitution of fuzzy sets and is not predicated upon a dichotomization of fuzzy membership scores. To illustrate these procedures, the same data set is used as in the previous chapters. However, the original interval-scale data are converted into fuzzy membership scores (which range from 0 to 1), thereby avoiding dichotomizing or trichotomizing the data (i.e., sorting the cases into crude categories). Of course, the important qualitative states of full membership (fuzzy membership = 1.0) and full nonmembership (fuzzy membership = 0.0) are retained, which makes fuzzy sets simultaneously qualitative and quantitative. It is important to point out that the analytic approach sketched in this chapter offers a new way to conduct fuzzy-set analysis of social data. This new [Page 89]analytic strategy is superior in several respects to the one sketched in Fuzzy-Set Social Science (Ragin, 2000). While both approaches have strengths and weaknesses, the one presented here uses the truth table as the key analytic device. As shall be demonstrated, a further advantage of the fuzzy-set truthtable approach is that it is more transparent. Thus, the researcher has more direct control over the process of data analysis. This type of control is central to the practice of case-oriented research.